Journal of Topology最新文献

We introduce a new method for proving twisted homological stability, and use it to prove such results for symmetric groups and general linear groups. In addition to sometimes slightly improving the stable range given by the traditional method (due to Dwyer), it is easier to adapt to nonstandard situations. As an illustration of this, we generalize to $�{GL}_n$ of many rings $�R$ a theorem of Borel that says that passing from $�{GL}_n$ of a number ring to a finite-index subgroup does not change the rational cohomology. Charney proved this generalization for trivial coefficients, and we extend it to twisted coefficients.

We introduce the notion of hyperbolic orientation of a motivic ring spectrum, which generalises the various existing notions of orientation (by the groups $�GL$, $�{SL}^c$, $�SL$, $�Sp$). We show that hyperbolic orientations of $�\eta$-periodic ring spectra correspond to theories of Pontryagin classes, much in the same way that $�GL$-orientations of arbitrary ring spectra correspond to theories of Chern classes. We prove that $�\eta$-periodic hyperbolically oriented cohomology theories do not admit further characteristic classes for vector bundles, by computing the cohomology of the étale classifying space $�{BGL}_n$. Finally, we construct the universal hyperbolically oriented $�\eta$-periodic commutative motivic ring spectrum, an analogue of Voevodsky's cobordism spectrum $�MGL$.

We establish motivic versions of the theorems of Lin and Gunawardena, thereby confirming the motivic Segal conjecture for the algebraic group $�{\mu }_{\ell }$ of $�\ell$th roots of unity, where $�\ell$ is any prime. To achieve this we develop motivic Singer constructions associated to the symmetric group $�S_{\ell }$ and to $�{\mu }_{\ell }$, and introduce a delayed limit Adams spectral sequence.

We study the homotopy of the connected sum of a manifold with a projective space, viewed as a typical way to stabilize manifolds. In particular, we show a loop homotopy decomposition of a manifold after stabilization by a projective space, and provide concrete examples. To do this, we trace the effect in homotopy theory of surgery on certain product manifolds by showing a loop homotopy decomposition after localization away from the order of the image of the classical $�J$-homomorphism.

We show that, for $��\epsilon �=�\frac{1}{4000}�$, any action of a finite cyclic group by $��(�1�+�\epsilon �)�$-bilipschitz homeomorphisms on a closed 3-manifold is conjugated to a smooth action.

We give a dynamical characterization of convex cocompact group actions on properly convex domains in projective space in the sense of Danciger–Guéritaud–Kassel: we show that convex cocompactness in $��R�P^d�$ is equivalent to an expansion property of the group about its limit set, occurring in different Grassmannians. As an application, we give a sufficient and necessary condition for convex cocompactness for groups that are hyperbolic relative to a collection of convex cocompact subgroups. We show that convex cocompactness in this situation is equivalent to the existence of an equivariant homeomorphism from the Bowditch boundary to the quotient of the limit set of the group by the limit sets of its peripheral subgroups.

In this paper we study the automorphism group of the procongruence mapping class group through its action on the associated procongruence curve and pants complexes. Our main result is a rigidity theorem for the procongruence completion of the pants complex. As an application we prove that moduli stacks of smooth algebraic curves satisfy a weak anabelian property in the procongruence setting.